Let us give the intuition why this bound is optimal, focusing on large and small scales only.
Our previous remark ensures that \(\inf _{S\subset F(Q_n)\ :\ \delta n <|S| \le n/2} \frac{|\partial S|^{4/3}}{|S|} \approx n^{-2/3+o(1)}\) , so the bound of Theorem REF  is indeed optimal for sets containing a macroscopic proportion of faces of \(Q_n\) .
For small scales, [1]} states that the supremum over all cycles of length 2 of the number of faces contained in the smallest component of the complement of the cycle is at most \(n^{2/3+o(1)}\) . We expect the supremum to be indeed \(n^{2/3+o(1)}\) ; if this is true, then the bound of Theorem REF  is also optimal for sets containing at most \(n^{2/3+o(1)}\)  faces.
